# Properties

 Label 175.2.a.f.1.1 Level $175$ Weight $2$ Character 175.1 Self dual yes Analytic conductor $1.397$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(1,175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 175.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155 q^{2} +2.56155 q^{3} +0.438447 q^{4} -4.00000 q^{6} +1.00000 q^{7} +2.43845 q^{8} +3.56155 q^{9} +O(q^{10})$$ $$q-1.56155 q^{2} +2.56155 q^{3} +0.438447 q^{4} -4.00000 q^{6} +1.00000 q^{7} +2.43845 q^{8} +3.56155 q^{9} +2.56155 q^{11} +1.12311 q^{12} -4.56155 q^{13} -1.56155 q^{14} -4.68466 q^{16} +4.56155 q^{17} -5.56155 q^{18} +1.12311 q^{19} +2.56155 q^{21} -4.00000 q^{22} +5.12311 q^{23} +6.24621 q^{24} +7.12311 q^{26} +1.43845 q^{27} +0.438447 q^{28} -5.68466 q^{29} +2.43845 q^{32} +6.56155 q^{33} -7.12311 q^{34} +1.56155 q^{36} -6.00000 q^{37} -1.75379 q^{38} -11.6847 q^{39} -3.12311 q^{41} -4.00000 q^{42} -9.12311 q^{43} +1.12311 q^{44} -8.00000 q^{46} -3.68466 q^{47} -12.0000 q^{48} +1.00000 q^{49} +11.6847 q^{51} -2.00000 q^{52} -3.12311 q^{53} -2.24621 q^{54} +2.43845 q^{56} +2.87689 q^{57} +8.87689 q^{58} -4.00000 q^{59} -9.36932 q^{61} +3.56155 q^{63} +5.56155 q^{64} -10.2462 q^{66} +6.24621 q^{67} +2.00000 q^{68} +13.1231 q^{69} +8.00000 q^{71} +8.68466 q^{72} -4.24621 q^{73} +9.36932 q^{74} +0.492423 q^{76} +2.56155 q^{77} +18.2462 q^{78} -6.56155 q^{79} -7.00000 q^{81} +4.87689 q^{82} -4.00000 q^{83} +1.12311 q^{84} +14.2462 q^{86} -14.5616 q^{87} +6.24621 q^{88} +7.12311 q^{89} -4.56155 q^{91} +2.24621 q^{92} +5.75379 q^{94} +6.24621 q^{96} +14.8078 q^{97} -1.56155 q^{98} +9.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 + 5 * q^4 - 8 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 $$2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + q^{11} - 6 q^{12} - 5 q^{13} + q^{14} + 3 q^{16} + 5 q^{17} - 7 q^{18} - 6 q^{19} + q^{21} - 8 q^{22} + 2 q^{23} - 4 q^{24} + 6 q^{26} + 7 q^{27} + 5 q^{28} + q^{29} + 9 q^{32} + 9 q^{33} - 6 q^{34} - q^{36} - 12 q^{37} - 20 q^{38} - 11 q^{39} + 2 q^{41} - 8 q^{42} - 10 q^{43} - 6 q^{44} - 16 q^{46} + 5 q^{47} - 24 q^{48} + 2 q^{49} + 11 q^{51} - 4 q^{52} + 2 q^{53} + 12 q^{54} + 9 q^{56} + 14 q^{57} + 26 q^{58} - 8 q^{59} + 6 q^{61} + 3 q^{63} + 7 q^{64} - 4 q^{66} - 4 q^{67} + 4 q^{68} + 18 q^{69} + 16 q^{71} + 5 q^{72} + 8 q^{73} - 6 q^{74} - 32 q^{76} + q^{77} + 20 q^{78} - 9 q^{79} - 14 q^{81} + 18 q^{82} - 8 q^{83} - 6 q^{84} + 12 q^{86} - 25 q^{87} - 4 q^{88} + 6 q^{89} - 5 q^{91} - 12 q^{92} + 28 q^{94} - 4 q^{96} + 9 q^{97} + q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 + 5 * q^4 - 8 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 + q^11 - 6 * q^12 - 5 * q^13 + q^14 + 3 * q^16 + 5 * q^17 - 7 * q^18 - 6 * q^19 + q^21 - 8 * q^22 + 2 * q^23 - 4 * q^24 + 6 * q^26 + 7 * q^27 + 5 * q^28 + q^29 + 9 * q^32 + 9 * q^33 - 6 * q^34 - q^36 - 12 * q^37 - 20 * q^38 - 11 * q^39 + 2 * q^41 - 8 * q^42 - 10 * q^43 - 6 * q^44 - 16 * q^46 + 5 * q^47 - 24 * q^48 + 2 * q^49 + 11 * q^51 - 4 * q^52 + 2 * q^53 + 12 * q^54 + 9 * q^56 + 14 * q^57 + 26 * q^58 - 8 * q^59 + 6 * q^61 + 3 * q^63 + 7 * q^64 - 4 * q^66 - 4 * q^67 + 4 * q^68 + 18 * q^69 + 16 * q^71 + 5 * q^72 + 8 * q^73 - 6 * q^74 - 32 * q^76 + q^77 + 20 * q^78 - 9 * q^79 - 14 * q^81 + 18 * q^82 - 8 * q^83 - 6 * q^84 + 12 * q^86 - 25 * q^87 - 4 * q^88 + 6 * q^89 - 5 * q^91 - 12 * q^92 + 28 * q^94 - 4 * q^96 + 9 * q^97 + q^98 + 10 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.56155 −1.10418 −0.552092 0.833783i $$-0.686170\pi$$
−0.552092 + 0.833783i $$0.686170\pi$$
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 0.438447 0.219224
$$5$$ 0 0
$$6$$ −4.00000 −1.63299
$$7$$ 1.00000 0.377964
$$8$$ 2.43845 0.862121
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ 2.56155 0.772337 0.386169 0.922428i $$-0.373798\pi$$
0.386169 + 0.922428i $$0.373798\pi$$
$$12$$ 1.12311 0.324213
$$13$$ −4.56155 −1.26515 −0.632574 0.774500i $$-0.718001\pi$$
−0.632574 + 0.774500i $$0.718001\pi$$
$$14$$ −1.56155 −0.417343
$$15$$ 0 0
$$16$$ −4.68466 −1.17116
$$17$$ 4.56155 1.10634 0.553170 0.833069i $$-0.313418\pi$$
0.553170 + 0.833069i $$0.313418\pi$$
$$18$$ −5.56155 −1.31087
$$19$$ 1.12311 0.257658 0.128829 0.991667i $$-0.458878\pi$$
0.128829 + 0.991667i $$0.458878\pi$$
$$20$$ 0 0
$$21$$ 2.56155 0.558977
$$22$$ −4.00000 −0.852803
$$23$$ 5.12311 1.06824 0.534121 0.845408i $$-0.320643\pi$$
0.534121 + 0.845408i $$0.320643\pi$$
$$24$$ 6.24621 1.27500
$$25$$ 0 0
$$26$$ 7.12311 1.39696
$$27$$ 1.43845 0.276829
$$28$$ 0.438447 0.0828587
$$29$$ −5.68466 −1.05561 −0.527807 0.849364i $$-0.676986\pi$$
−0.527807 + 0.849364i $$0.676986\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 2.43845 0.431061
$$33$$ 6.56155 1.14222
$$34$$ −7.12311 −1.22160
$$35$$ 0 0
$$36$$ 1.56155 0.260259
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −1.75379 −0.284502
$$39$$ −11.6847 −1.87104
$$40$$ 0 0
$$41$$ −3.12311 −0.487747 −0.243874 0.969807i $$-0.578418\pi$$
−0.243874 + 0.969807i $$0.578418\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −9.12311 −1.39126 −0.695630 0.718400i $$-0.744875\pi$$
−0.695630 + 0.718400i $$0.744875\pi$$
$$44$$ 1.12311 0.169315
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ −3.68466 −0.537463 −0.268731 0.963215i $$-0.586604\pi$$
−0.268731 + 0.963215i $$0.586604\pi$$
$$48$$ −12.0000 −1.73205
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 11.6847 1.63618
$$52$$ −2.00000 −0.277350
$$53$$ −3.12311 −0.428992 −0.214496 0.976725i $$-0.568811\pi$$
−0.214496 + 0.976725i $$0.568811\pi$$
$$54$$ −2.24621 −0.305671
$$55$$ 0 0
$$56$$ 2.43845 0.325851
$$57$$ 2.87689 0.381054
$$58$$ 8.87689 1.16559
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −9.36932 −1.19962 −0.599809 0.800143i $$-0.704757\pi$$
−0.599809 + 0.800143i $$0.704757\pi$$
$$62$$ 0 0
$$63$$ 3.56155 0.448713
$$64$$ 5.56155 0.695194
$$65$$ 0 0
$$66$$ −10.2462 −1.26122
$$67$$ 6.24621 0.763096 0.381548 0.924349i $$-0.375391\pi$$
0.381548 + 0.924349i $$0.375391\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 13.1231 1.57984
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 8.68466 1.02350
$$73$$ −4.24621 −0.496981 −0.248491 0.968634i $$-0.579935\pi$$
−0.248491 + 0.968634i $$0.579935\pi$$
$$74$$ 9.36932 1.08916
$$75$$ 0 0
$$76$$ 0.492423 0.0564847
$$77$$ 2.56155 0.291916
$$78$$ 18.2462 2.06598
$$79$$ −6.56155 −0.738232 −0.369116 0.929383i $$-0.620340\pi$$
−0.369116 + 0.929383i $$0.620340\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 4.87689 0.538563
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 1.12311 0.122541
$$85$$ 0 0
$$86$$ 14.2462 1.53621
$$87$$ −14.5616 −1.56116
$$88$$ 6.24621 0.665848
$$89$$ 7.12311 0.755048 0.377524 0.926000i $$-0.376776\pi$$
0.377524 + 0.926000i $$0.376776\pi$$
$$90$$ 0 0
$$91$$ −4.56155 −0.478181
$$92$$ 2.24621 0.234184
$$93$$ 0 0
$$94$$ 5.75379 0.593458
$$95$$ 0 0
$$96$$ 6.24621 0.637501
$$97$$ 14.8078 1.50350 0.751750 0.659448i $$-0.229210\pi$$
0.751750 + 0.659448i $$0.229210\pi$$
$$98$$ −1.56155 −0.157741
$$99$$ 9.12311 0.916907
$$100$$ 0 0
$$101$$ 0.246211 0.0244989 0.0122495 0.999925i $$-0.496101\pi$$
0.0122495 + 0.999925i $$0.496101\pi$$
$$102$$ −18.2462 −1.80664
$$103$$ −1.43845 −0.141734 −0.0708672 0.997486i $$-0.522577\pi$$
−0.0708672 + 0.997486i $$0.522577\pi$$
$$104$$ −11.1231 −1.09071
$$105$$ 0 0
$$106$$ 4.87689 0.473686
$$107$$ 11.3693 1.09911 0.549557 0.835456i $$-0.314797\pi$$
0.549557 + 0.835456i $$0.314797\pi$$
$$108$$ 0.630683 0.0606875
$$109$$ 17.6847 1.69388 0.846942 0.531686i $$-0.178441\pi$$
0.846942 + 0.531686i $$0.178441\pi$$
$$110$$ 0 0
$$111$$ −15.3693 −1.45879
$$112$$ −4.68466 −0.442659
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ −4.49242 −0.420754
$$115$$ 0 0
$$116$$ −2.49242 −0.231416
$$117$$ −16.2462 −1.50196
$$118$$ 6.24621 0.575010
$$119$$ 4.56155 0.418157
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ 14.6307 1.32460
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −5.56155 −0.495463
$$127$$ −10.2462 −0.909204 −0.454602 0.890695i $$-0.650219\pi$$
−0.454602 + 0.890695i $$0.650219\pi$$
$$128$$ −13.5616 −1.19868
$$129$$ −23.3693 −2.05755
$$130$$ 0 0
$$131$$ −9.12311 −0.797089 −0.398545 0.917149i $$-0.630485\pi$$
−0.398545 + 0.917149i $$0.630485\pi$$
$$132$$ 2.87689 0.250402
$$133$$ 1.12311 0.0973856
$$134$$ −9.75379 −0.842599
$$135$$ 0 0
$$136$$ 11.1231 0.953798
$$137$$ 8.87689 0.758404 0.379202 0.925314i $$-0.376199\pi$$
0.379202 + 0.925314i $$0.376199\pi$$
$$138$$ −20.4924 −1.74443
$$139$$ −6.87689 −0.583291 −0.291645 0.956527i $$-0.594203\pi$$
−0.291645 + 0.956527i $$0.594203\pi$$
$$140$$ 0 0
$$141$$ −9.43845 −0.794861
$$142$$ −12.4924 −1.04834
$$143$$ −11.6847 −0.977120
$$144$$ −16.6847 −1.39039
$$145$$ 0 0
$$146$$ 6.63068 0.548759
$$147$$ 2.56155 0.211273
$$148$$ −2.63068 −0.216241
$$149$$ −4.24621 −0.347863 −0.173932 0.984758i $$-0.555647\pi$$
−0.173932 + 0.984758i $$0.555647\pi$$
$$150$$ 0 0
$$151$$ 21.9309 1.78471 0.892354 0.451335i $$-0.149052\pi$$
0.892354 + 0.451335i $$0.149052\pi$$
$$152$$ 2.73863 0.222133
$$153$$ 16.2462 1.31343
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ −5.12311 −0.410177
$$157$$ −3.75379 −0.299585 −0.149792 0.988717i $$-0.547861\pi$$
−0.149792 + 0.988717i $$0.547861\pi$$
$$158$$ 10.2462 0.815145
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 5.12311 0.403757
$$162$$ 10.9309 0.858810
$$163$$ −1.12311 −0.0879684 −0.0439842 0.999032i $$-0.514005\pi$$
−0.0439842 + 0.999032i $$0.514005\pi$$
$$164$$ −1.36932 −0.106926
$$165$$ 0 0
$$166$$ 6.24621 0.484800
$$167$$ −21.9309 −1.69706 −0.848531 0.529146i $$-0.822512\pi$$
−0.848531 + 0.529146i $$0.822512\pi$$
$$168$$ 6.24621 0.481906
$$169$$ 7.80776 0.600597
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 8.56155 0.650923 0.325461 0.945555i $$-0.394480\pi$$
0.325461 + 0.945555i $$0.394480\pi$$
$$174$$ 22.7386 1.72381
$$175$$ 0 0
$$176$$ −12.0000 −0.904534
$$177$$ −10.2462 −0.770152
$$178$$ −11.1231 −0.833712
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 23.6155 1.75533 0.877664 0.479276i $$-0.159101\pi$$
0.877664 + 0.479276i $$0.159101\pi$$
$$182$$ 7.12311 0.528000
$$183$$ −24.0000 −1.77413
$$184$$ 12.4924 0.920954
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 11.6847 0.854467
$$188$$ −1.61553 −0.117824
$$189$$ 1.43845 0.104632
$$190$$ 0 0
$$191$$ −9.43845 −0.682942 −0.341471 0.939892i $$-0.610925\pi$$
−0.341471 + 0.939892i $$0.610925\pi$$
$$192$$ 14.2462 1.02813
$$193$$ 5.36932 0.386492 0.193246 0.981150i $$-0.438098\pi$$
0.193246 + 0.981150i $$0.438098\pi$$
$$194$$ −23.1231 −1.66014
$$195$$ 0 0
$$196$$ 0.438447 0.0313177
$$197$$ 7.12311 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$198$$ −14.2462 −1.01243
$$199$$ −18.2462 −1.29344 −0.646720 0.762728i $$-0.723860\pi$$
−0.646720 + 0.762728i $$0.723860\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ −0.384472 −0.0270513
$$203$$ −5.68466 −0.398985
$$204$$ 5.12311 0.358689
$$205$$ 0 0
$$206$$ 2.24621 0.156501
$$207$$ 18.2462 1.26820
$$208$$ 21.3693 1.48170
$$209$$ 2.87689 0.198999
$$210$$ 0 0
$$211$$ −23.0540 −1.58710 −0.793551 0.608504i $$-0.791770\pi$$
−0.793551 + 0.608504i $$0.791770\pi$$
$$212$$ −1.36932 −0.0940451
$$213$$ 20.4924 1.40412
$$214$$ −17.7538 −1.21362
$$215$$ 0 0
$$216$$ 3.50758 0.238660
$$217$$ 0 0
$$218$$ −27.6155 −1.87036
$$219$$ −10.8769 −0.734992
$$220$$ 0 0
$$221$$ −20.8078 −1.39968
$$222$$ 24.0000 1.61077
$$223$$ 6.56155 0.439394 0.219697 0.975568i $$-0.429493\pi$$
0.219697 + 0.975568i $$0.429493\pi$$
$$224$$ 2.43845 0.162926
$$225$$ 0 0
$$226$$ −21.8617 −1.45422
$$227$$ −23.6847 −1.57201 −0.786003 0.618223i $$-0.787853\pi$$
−0.786003 + 0.618223i $$0.787853\pi$$
$$228$$ 1.26137 0.0835360
$$229$$ 19.1231 1.26369 0.631845 0.775095i $$-0.282298\pi$$
0.631845 + 0.775095i $$0.282298\pi$$
$$230$$ 0 0
$$231$$ 6.56155 0.431718
$$232$$ −13.8617 −0.910068
$$233$$ 3.12311 0.204601 0.102301 0.994754i $$-0.467380\pi$$
0.102301 + 0.994754i $$0.467380\pi$$
$$234$$ 25.3693 1.65844
$$235$$ 0 0
$$236$$ −1.75379 −0.114162
$$237$$ −16.8078 −1.09178
$$238$$ −7.12311 −0.461722
$$239$$ −0.807764 −0.0522499 −0.0261250 0.999659i $$-0.508317\pi$$
−0.0261250 + 0.999659i $$0.508317\pi$$
$$240$$ 0 0
$$241$$ 12.2462 0.788848 0.394424 0.918929i $$-0.370944\pi$$
0.394424 + 0.918929i $$0.370944\pi$$
$$242$$ 6.93087 0.445533
$$243$$ −22.2462 −1.42710
$$244$$ −4.10795 −0.262985
$$245$$ 0 0
$$246$$ 12.4924 0.796488
$$247$$ −5.12311 −0.325975
$$248$$ 0 0
$$249$$ −10.2462 −0.649327
$$250$$ 0 0
$$251$$ −17.1231 −1.08080 −0.540400 0.841408i $$-0.681727\pi$$
−0.540400 + 0.841408i $$0.681727\pi$$
$$252$$ 1.56155 0.0983686
$$253$$ 13.1231 0.825043
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 10.0540 0.628373
$$257$$ −22.4924 −1.40304 −0.701519 0.712650i $$-0.747495\pi$$
−0.701519 + 0.712650i $$0.747495\pi$$
$$258$$ 36.4924 2.27192
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −20.2462 −1.25321
$$262$$ 14.2462 0.880134
$$263$$ 21.1231 1.30251 0.651253 0.758860i $$-0.274244\pi$$
0.651253 + 0.758860i $$0.274244\pi$$
$$264$$ 16.0000 0.984732
$$265$$ 0 0
$$266$$ −1.75379 −0.107532
$$267$$ 18.2462 1.11665
$$268$$ 2.73863 0.167289
$$269$$ 28.7386 1.75223 0.876113 0.482106i $$-0.160128\pi$$
0.876113 + 0.482106i $$0.160128\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −21.3693 −1.29571
$$273$$ −11.6847 −0.707188
$$274$$ −13.8617 −0.837418
$$275$$ 0 0
$$276$$ 5.75379 0.346337
$$277$$ −16.2462 −0.976140 −0.488070 0.872804i $$-0.662299\pi$$
−0.488070 + 0.872804i $$0.662299\pi$$
$$278$$ 10.7386 0.644060
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.5616 0.987979 0.493990 0.869468i $$-0.335538\pi$$
0.493990 + 0.869468i $$0.335538\pi$$
$$282$$ 14.7386 0.877673
$$283$$ 23.6847 1.40791 0.703953 0.710246i $$-0.251416\pi$$
0.703953 + 0.710246i $$0.251416\pi$$
$$284$$ 3.50758 0.208136
$$285$$ 0 0
$$286$$ 18.2462 1.07892
$$287$$ −3.12311 −0.184351
$$288$$ 8.68466 0.511748
$$289$$ 3.80776 0.223986
$$290$$ 0 0
$$291$$ 37.9309 2.22355
$$292$$ −1.86174 −0.108950
$$293$$ −9.68466 −0.565784 −0.282892 0.959152i $$-0.591294\pi$$
−0.282892 + 0.959152i $$0.591294\pi$$
$$294$$ −4.00000 −0.233285
$$295$$ 0 0
$$296$$ −14.6307 −0.850391
$$297$$ 3.68466 0.213806
$$298$$ 6.63068 0.384105
$$299$$ −23.3693 −1.35148
$$300$$ 0 0
$$301$$ −9.12311 −0.525847
$$302$$ −34.2462 −1.97065
$$303$$ 0.630683 0.0362318
$$304$$ −5.26137 −0.301760
$$305$$ 0 0
$$306$$ −25.3693 −1.45027
$$307$$ 31.6847 1.80834 0.904169 0.427174i $$-0.140491\pi$$
0.904169 + 0.427174i $$0.140491\pi$$
$$308$$ 1.12311 0.0639949
$$309$$ −3.68466 −0.209613
$$310$$ 0 0
$$311$$ −9.61553 −0.545247 −0.272623 0.962121i $$-0.587891\pi$$
−0.272623 + 0.962121i $$0.587891\pi$$
$$312$$ −28.4924 −1.61307
$$313$$ −31.3002 −1.76919 −0.884596 0.466359i $$-0.845566\pi$$
−0.884596 + 0.466359i $$0.845566\pi$$
$$314$$ 5.86174 0.330797
$$315$$ 0 0
$$316$$ −2.87689 −0.161838
$$317$$ 22.4924 1.26330 0.631650 0.775254i $$-0.282378\pi$$
0.631650 + 0.775254i $$0.282378\pi$$
$$318$$ 12.4924 0.700540
$$319$$ −14.5616 −0.815290
$$320$$ 0 0
$$321$$ 29.1231 1.62549
$$322$$ −8.00000 −0.445823
$$323$$ 5.12311 0.285057
$$324$$ −3.06913 −0.170507
$$325$$ 0 0
$$326$$ 1.75379 0.0971334
$$327$$ 45.3002 2.50511
$$328$$ −7.61553 −0.420497
$$329$$ −3.68466 −0.203142
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −1.75379 −0.0962517
$$333$$ −21.3693 −1.17103
$$334$$ 34.2462 1.87387
$$335$$ 0 0
$$336$$ −12.0000 −0.654654
$$337$$ 34.4924 1.87892 0.939461 0.342656i $$-0.111326\pi$$
0.939461 + 0.342656i $$0.111326\pi$$
$$338$$ −12.1922 −0.663170
$$339$$ 35.8617 1.94774
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −6.24621 −0.337756
$$343$$ 1.00000 0.0539949
$$344$$ −22.2462 −1.19944
$$345$$ 0 0
$$346$$ −13.3693 −0.718739
$$347$$ 1.12311 0.0602915 0.0301457 0.999546i $$-0.490403\pi$$
0.0301457 + 0.999546i $$0.490403\pi$$
$$348$$ −6.38447 −0.342244
$$349$$ −22.4924 −1.20399 −0.601996 0.798499i $$-0.705628\pi$$
−0.601996 + 0.798499i $$0.705628\pi$$
$$350$$ 0 0
$$351$$ −6.56155 −0.350230
$$352$$ 6.24621 0.332924
$$353$$ 14.8078 0.788138 0.394069 0.919081i $$-0.371067\pi$$
0.394069 + 0.919081i $$0.371067\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 0 0
$$356$$ 3.12311 0.165524
$$357$$ 11.6847 0.618418
$$358$$ −31.2311 −1.65061
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −17.7386 −0.933612
$$362$$ −36.8769 −1.93821
$$363$$ −11.3693 −0.596734
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 37.4773 1.95897
$$367$$ −3.68466 −0.192338 −0.0961688 0.995365i $$-0.530659\pi$$
−0.0961688 + 0.995365i $$0.530659\pi$$
$$368$$ −24.0000 −1.25109
$$369$$ −11.1231 −0.579046
$$370$$ 0 0
$$371$$ −3.12311 −0.162144
$$372$$ 0 0
$$373$$ −29.3693 −1.52069 −0.760343 0.649522i $$-0.774969\pi$$
−0.760343 + 0.649522i $$0.774969\pi$$
$$374$$ −18.2462 −0.943489
$$375$$ 0 0
$$376$$ −8.98485 −0.463358
$$377$$ 25.9309 1.33551
$$378$$ −2.24621 −0.115533
$$379$$ 16.4924 0.847159 0.423579 0.905859i $$-0.360773\pi$$
0.423579 + 0.905859i $$0.360773\pi$$
$$380$$ 0 0
$$381$$ −26.2462 −1.34463
$$382$$ 14.7386 0.754094
$$383$$ 10.2462 0.523557 0.261778 0.965128i $$-0.415691\pi$$
0.261778 + 0.965128i $$0.415691\pi$$
$$384$$ −34.7386 −1.77275
$$385$$ 0 0
$$386$$ −8.38447 −0.426758
$$387$$ −32.4924 −1.65168
$$388$$ 6.49242 0.329603
$$389$$ 3.93087 0.199303 0.0996515 0.995022i $$-0.468227\pi$$
0.0996515 + 0.995022i $$0.468227\pi$$
$$390$$ 0 0
$$391$$ 23.3693 1.18184
$$392$$ 2.43845 0.123160
$$393$$ −23.3693 −1.17883
$$394$$ −11.1231 −0.560374
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ −23.4384 −1.17634 −0.588171 0.808737i $$-0.700152\pi$$
−0.588171 + 0.808737i $$0.700152\pi$$
$$398$$ 28.4924 1.42820
$$399$$ 2.87689 0.144025
$$400$$ 0 0
$$401$$ 27.4384 1.37021 0.685105 0.728444i $$-0.259756\pi$$
0.685105 + 0.728444i $$0.259756\pi$$
$$402$$ −24.9848 −1.24613
$$403$$ 0 0
$$404$$ 0.107951 0.00537074
$$405$$ 0 0
$$406$$ 8.87689 0.440553
$$407$$ −15.3693 −0.761829
$$408$$ 28.4924 1.41059
$$409$$ −26.4924 −1.30997 −0.654983 0.755644i $$-0.727324\pi$$
−0.654983 + 0.755644i $$0.727324\pi$$
$$410$$ 0 0
$$411$$ 22.7386 1.12161
$$412$$ −0.630683 −0.0310715
$$413$$ −4.00000 −0.196827
$$414$$ −28.4924 −1.40033
$$415$$ 0 0
$$416$$ −11.1231 −0.545355
$$417$$ −17.6155 −0.862636
$$418$$ −4.49242 −0.219732
$$419$$ 9.75379 0.476504 0.238252 0.971203i $$-0.423426\pi$$
0.238252 + 0.971203i $$0.423426\pi$$
$$420$$ 0 0
$$421$$ 9.68466 0.472001 0.236001 0.971753i $$-0.424163\pi$$
0.236001 + 0.971753i $$0.424163\pi$$
$$422$$ 36.0000 1.75245
$$423$$ −13.1231 −0.638067
$$424$$ −7.61553 −0.369843
$$425$$ 0 0
$$426$$ −32.0000 −1.55041
$$427$$ −9.36932 −0.453413
$$428$$ 4.98485 0.240952
$$429$$ −29.9309 −1.44508
$$430$$ 0 0
$$431$$ 0.807764 0.0389086 0.0194543 0.999811i $$-0.493807\pi$$
0.0194543 + 0.999811i $$0.493807\pi$$
$$432$$ −6.73863 −0.324213
$$433$$ 8.24621 0.396288 0.198144 0.980173i $$-0.436509\pi$$
0.198144 + 0.980173i $$0.436509\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7.75379 0.371339
$$437$$ 5.75379 0.275241
$$438$$ 16.9848 0.811567
$$439$$ −15.3693 −0.733537 −0.366769 0.930312i $$-0.619536\pi$$
−0.366769 + 0.930312i $$0.619536\pi$$
$$440$$ 0 0
$$441$$ 3.56155 0.169598
$$442$$ 32.4924 1.54551
$$443$$ 27.3693 1.30036 0.650178 0.759782i $$-0.274694\pi$$
0.650178 + 0.759782i $$0.274694\pi$$
$$444$$ −6.73863 −0.319801
$$445$$ 0 0
$$446$$ −10.2462 −0.485172
$$447$$ −10.8769 −0.514459
$$448$$ 5.56155 0.262759
$$449$$ 18.8078 0.887593 0.443797 0.896128i $$-0.353631\pi$$
0.443797 + 0.896128i $$0.353631\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 6.13826 0.288719
$$453$$ 56.1771 2.63943
$$454$$ 36.9848 1.73578
$$455$$ 0 0
$$456$$ 7.01515 0.328515
$$457$$ 8.87689 0.415244 0.207622 0.978209i $$-0.433428\pi$$
0.207622 + 0.978209i $$0.433428\pi$$
$$458$$ −29.8617 −1.39535
$$459$$ 6.56155 0.306267
$$460$$ 0 0
$$461$$ −4.87689 −0.227140 −0.113570 0.993530i $$-0.536229\pi$$
−0.113570 + 0.993530i $$0.536229\pi$$
$$462$$ −10.2462 −0.476697
$$463$$ 20.4924 0.952364 0.476182 0.879347i $$-0.342020\pi$$
0.476182 + 0.879347i $$0.342020\pi$$
$$464$$ 26.6307 1.23630
$$465$$ 0 0
$$466$$ −4.87689 −0.225918
$$467$$ −26.5616 −1.22912 −0.614561 0.788869i $$-0.710667\pi$$
−0.614561 + 0.788869i $$0.710667\pi$$
$$468$$ −7.12311 −0.329266
$$469$$ 6.24621 0.288423
$$470$$ 0 0
$$471$$ −9.61553 −0.443060
$$472$$ −9.75379 −0.448955
$$473$$ −23.3693 −1.07452
$$474$$ 26.2462 1.20553
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ −11.1231 −0.509292
$$478$$ 1.26137 0.0576935
$$479$$ 13.1231 0.599610 0.299805 0.954001i $$-0.403078\pi$$
0.299805 + 0.954001i $$0.403078\pi$$
$$480$$ 0 0
$$481$$ 27.3693 1.24793
$$482$$ −19.1231 −0.871034
$$483$$ 13.1231 0.597122
$$484$$ −1.94602 −0.0884557
$$485$$ 0 0
$$486$$ 34.7386 1.57578
$$487$$ −5.12311 −0.232150 −0.116075 0.993240i $$-0.537031\pi$$
−0.116075 + 0.993240i $$0.537031\pi$$
$$488$$ −22.8466 −1.03422
$$489$$ −2.87689 −0.130098
$$490$$ 0 0
$$491$$ 4.17708 0.188509 0.0942545 0.995548i $$-0.469953\pi$$
0.0942545 + 0.995548i $$0.469953\pi$$
$$492$$ −3.50758 −0.158134
$$493$$ −25.9309 −1.16787
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000 0.358849
$$498$$ 16.0000 0.716977
$$499$$ −4.17708 −0.186992 −0.0934959 0.995620i $$-0.529804\pi$$
−0.0934959 + 0.995620i $$0.529804\pi$$
$$500$$ 0 0
$$501$$ −56.1771 −2.50981
$$502$$ 26.7386 1.19340
$$503$$ −10.0691 −0.448960 −0.224480 0.974479i $$-0.572068\pi$$
−0.224480 + 0.974479i $$0.572068\pi$$
$$504$$ 8.68466 0.386845
$$505$$ 0 0
$$506$$ −20.4924 −0.910999
$$507$$ 20.0000 0.888231
$$508$$ −4.49242 −0.199319
$$509$$ −28.2462 −1.25199 −0.625996 0.779827i $$-0.715307\pi$$
−0.625996 + 0.779827i $$0.715307\pi$$
$$510$$ 0 0
$$511$$ −4.24621 −0.187841
$$512$$ 11.4233 0.504843
$$513$$ 1.61553 0.0713273
$$514$$ 35.1231 1.54921
$$515$$ 0 0
$$516$$ −10.2462 −0.451064
$$517$$ −9.43845 −0.415102
$$518$$ 9.36932 0.411664
$$519$$ 21.9309 0.962658
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 31.6155 1.38377
$$523$$ −7.50758 −0.328283 −0.164142 0.986437i $$-0.552485\pi$$
−0.164142 + 0.986437i $$0.552485\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ −32.9848 −1.43821
$$527$$ 0 0
$$528$$ −30.7386 −1.33773
$$529$$ 3.24621 0.141140
$$530$$ 0 0
$$531$$ −14.2462 −0.618233
$$532$$ 0.492423 0.0213492
$$533$$ 14.2462 0.617072
$$534$$ −28.4924 −1.23299
$$535$$ 0 0
$$536$$ 15.2311 0.657881
$$537$$ 51.2311 2.21078
$$538$$ −44.8769 −1.93478
$$539$$ 2.56155 0.110334
$$540$$ 0 0
$$541$$ −17.1922 −0.739152 −0.369576 0.929201i $$-0.620497\pi$$
−0.369576 + 0.929201i $$0.620497\pi$$
$$542$$ 24.9848 1.07319
$$543$$ 60.4924 2.59598
$$544$$ 11.1231 0.476899
$$545$$ 0 0
$$546$$ 18.2462 0.780866
$$547$$ −14.2462 −0.609124 −0.304562 0.952493i $$-0.598510\pi$$
−0.304562 + 0.952493i $$0.598510\pi$$
$$548$$ 3.89205 0.166260
$$549$$ −33.3693 −1.42417
$$550$$ 0 0
$$551$$ −6.38447 −0.271988
$$552$$ 32.0000 1.36201
$$553$$ −6.56155 −0.279026
$$554$$ 25.3693 1.07784
$$555$$ 0 0
$$556$$ −3.01515 −0.127871
$$557$$ 4.87689 0.206641 0.103320 0.994648i $$-0.467053\pi$$
0.103320 + 0.994648i $$0.467053\pi$$
$$558$$ 0 0
$$559$$ 41.6155 1.76015
$$560$$ 0 0
$$561$$ 29.9309 1.26368
$$562$$ −25.8617 −1.09091
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ −4.13826 −0.174252
$$565$$ 0 0
$$566$$ −36.9848 −1.55459
$$567$$ −7.00000 −0.293972
$$568$$ 19.5076 0.818520
$$569$$ 34.9848 1.46664 0.733320 0.679883i $$-0.237969\pi$$
0.733320 + 0.679883i $$0.237969\pi$$
$$570$$ 0 0
$$571$$ 7.50758 0.314182 0.157091 0.987584i $$-0.449788\pi$$
0.157091 + 0.987584i $$0.449788\pi$$
$$572$$ −5.12311 −0.214208
$$573$$ −24.1771 −1.01001
$$574$$ 4.87689 0.203558
$$575$$ 0 0
$$576$$ 19.8078 0.825324
$$577$$ −13.0540 −0.543444 −0.271722 0.962376i $$-0.587593\pi$$
−0.271722 + 0.962376i $$0.587593\pi$$
$$578$$ −5.94602 −0.247322
$$579$$ 13.7538 0.571588
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ −59.2311 −2.45521
$$583$$ −8.00000 −0.331326
$$584$$ −10.3542 −0.428458
$$585$$ 0 0
$$586$$ 15.1231 0.624730
$$587$$ 9.75379 0.402582 0.201291 0.979531i $$-0.435486\pi$$
0.201291 + 0.979531i $$0.435486\pi$$
$$588$$ 1.12311 0.0463161
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 18.2462 0.750549
$$592$$ 28.1080 1.15523
$$593$$ 23.4384 0.962502 0.481251 0.876583i $$-0.340183\pi$$
0.481251 + 0.876583i $$0.340183\pi$$
$$594$$ −5.75379 −0.236081
$$595$$ 0 0
$$596$$ −1.86174 −0.0762598
$$597$$ −46.7386 −1.91288
$$598$$ 36.4924 1.49229
$$599$$ 8.80776 0.359875 0.179938 0.983678i $$-0.442410\pi$$
0.179938 + 0.983678i $$0.442410\pi$$
$$600$$ 0 0
$$601$$ −26.4924 −1.08065 −0.540324 0.841457i $$-0.681698\pi$$
−0.540324 + 0.841457i $$0.681698\pi$$
$$602$$ 14.2462 0.580632
$$603$$ 22.2462 0.905936
$$604$$ 9.61553 0.391250
$$605$$ 0 0
$$606$$ −0.984845 −0.0400066
$$607$$ 4.94602 0.200753 0.100376 0.994950i $$-0.467995\pi$$
0.100376 + 0.994950i $$0.467995\pi$$
$$608$$ 2.73863 0.111066
$$609$$ −14.5616 −0.590064
$$610$$ 0 0
$$611$$ 16.8078 0.679969
$$612$$ 7.12311 0.287934
$$613$$ 8.73863 0.352950 0.176475 0.984305i $$-0.443531\pi$$
0.176475 + 0.984305i $$0.443531\pi$$
$$614$$ −49.4773 −1.99674
$$615$$ 0 0
$$616$$ 6.24621 0.251667
$$617$$ −15.7538 −0.634224 −0.317112 0.948388i $$-0.602713\pi$$
−0.317112 + 0.948388i $$0.602713\pi$$
$$618$$ 5.75379 0.231451
$$619$$ −42.1080 −1.69246 −0.846231 0.532817i $$-0.821134\pi$$
−0.846231 + 0.532817i $$0.821134\pi$$
$$620$$ 0 0
$$621$$ 7.36932 0.295720
$$622$$ 15.0152 0.602053
$$623$$ 7.12311 0.285381
$$624$$ 54.7386 2.19130
$$625$$ 0 0
$$626$$ 48.8769 1.95351
$$627$$ 7.36932 0.294302
$$628$$ −1.64584 −0.0656761
$$629$$ −27.3693 −1.09129
$$630$$ 0 0
$$631$$ 8.80776 0.350632 0.175316 0.984512i $$-0.443905\pi$$
0.175316 + 0.984512i $$0.443905\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ −59.0540 −2.34718
$$634$$ −35.1231 −1.39492
$$635$$ 0 0
$$636$$ −3.50758 −0.139084
$$637$$ −4.56155 −0.180735
$$638$$ 22.7386 0.900231
$$639$$ 28.4924 1.12714
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ −45.4773 −1.79484
$$643$$ 2.56155 0.101018 0.0505089 0.998724i $$-0.483916\pi$$
0.0505089 + 0.998724i $$0.483916\pi$$
$$644$$ 2.24621 0.0885131
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ −3.50758 −0.137897 −0.0689486 0.997620i $$-0.521964\pi$$
−0.0689486 + 0.997620i $$0.521964\pi$$
$$648$$ −17.0691 −0.670539
$$649$$ −10.2462 −0.402199
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −0.492423 −0.0192848
$$653$$ −49.2311 −1.92656 −0.963280 0.268499i $$-0.913473\pi$$
−0.963280 + 0.268499i $$0.913473\pi$$
$$654$$ −70.7386 −2.76610
$$655$$ 0 0
$$656$$ 14.6307 0.571232
$$657$$ −15.1231 −0.590009
$$658$$ 5.75379 0.224306
$$659$$ −36.1771 −1.40926 −0.704629 0.709575i $$-0.748887\pi$$
−0.704629 + 0.709575i $$0.748887\pi$$
$$660$$ 0 0
$$661$$ 3.12311 0.121475 0.0607374 0.998154i $$-0.480655\pi$$
0.0607374 + 0.998154i $$0.480655\pi$$
$$662$$ −18.7386 −0.728298
$$663$$ −53.3002 −2.07001
$$664$$ −9.75379 −0.378520
$$665$$ 0 0
$$666$$ 33.3693 1.29303
$$667$$ −29.1231 −1.12765
$$668$$ −9.61553 −0.372036
$$669$$ 16.8078 0.649826
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 6.24621 0.240953
$$673$$ 25.8617 0.996897 0.498448 0.866919i $$-0.333903\pi$$
0.498448 + 0.866919i $$0.333903\pi$$
$$674$$ −53.8617 −2.07468
$$675$$ 0 0
$$676$$ 3.42329 0.131665
$$677$$ 23.9309 0.919738 0.459869 0.887987i $$-0.347896\pi$$
0.459869 + 0.887987i $$0.347896\pi$$
$$678$$ −56.0000 −2.15067
$$679$$ 14.8078 0.568270
$$680$$ 0 0
$$681$$ −60.6695 −2.32486
$$682$$ 0 0
$$683$$ −42.7386 −1.63535 −0.817674 0.575681i $$-0.804737\pi$$
−0.817674 + 0.575681i $$0.804737\pi$$
$$684$$ 1.75379 0.0670578
$$685$$ 0 0
$$686$$ −1.56155 −0.0596204
$$687$$ 48.9848 1.86889
$$688$$ 42.7386 1.62940
$$689$$ 14.2462 0.542737
$$690$$ 0 0
$$691$$ 8.49242 0.323067 0.161533 0.986867i $$-0.448356\pi$$
0.161533 + 0.986867i $$0.448356\pi$$
$$692$$ 3.75379 0.142698
$$693$$ 9.12311 0.346558
$$694$$ −1.75379 −0.0665729
$$695$$ 0 0
$$696$$ −35.5076 −1.34591
$$697$$ −14.2462 −0.539614
$$698$$ 35.1231 1.32943
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ 0.0691303 0.00261102 0.00130551 0.999999i $$-0.499584\pi$$
0.00130551 + 0.999999i $$0.499584\pi$$
$$702$$ 10.2462 0.386718
$$703$$ −6.73863 −0.254152
$$704$$ 14.2462 0.536924
$$705$$ 0 0
$$706$$ −23.1231 −0.870250
$$707$$ 0.246211 0.00925973
$$708$$ −4.49242 −0.168836
$$709$$ −18.1771 −0.682655 −0.341327 0.939945i $$-0.610876\pi$$
−0.341327 + 0.939945i $$0.610876\pi$$
$$710$$ 0 0
$$711$$ −23.3693 −0.876418
$$712$$ 17.3693 0.650943
$$713$$ 0 0
$$714$$ −18.2462 −0.682847
$$715$$ 0 0
$$716$$ 8.76894 0.327711
$$717$$ −2.06913 −0.0772731
$$718$$ −12.4924 −0.466213
$$719$$ 49.6155 1.85035 0.925173 0.379544i $$-0.123919\pi$$
0.925173 + 0.379544i $$0.123919\pi$$
$$720$$ 0 0
$$721$$ −1.43845 −0.0535706
$$722$$ 27.6998 1.03088
$$723$$ 31.3693 1.16664
$$724$$ 10.3542 0.384809
$$725$$ 0 0
$$726$$ 17.7538 0.658905
$$727$$ −19.5076 −0.723496 −0.361748 0.932276i $$-0.617820\pi$$
−0.361748 + 0.932276i $$0.617820\pi$$
$$728$$ −11.1231 −0.412250
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ −41.6155 −1.53921
$$732$$ −10.5227 −0.388931
$$733$$ 5.68466 0.209968 0.104984 0.994474i $$-0.466521\pi$$
0.104984 + 0.994474i $$0.466521\pi$$
$$734$$ 5.75379 0.212376
$$735$$ 0 0
$$736$$ 12.4924 0.460477
$$737$$ 16.0000 0.589368
$$738$$ 17.3693 0.639373
$$739$$ 6.06913 0.223257 0.111628 0.993750i $$-0.464393\pi$$
0.111628 + 0.993750i $$0.464393\pi$$
$$740$$ 0 0
$$741$$ −13.1231 −0.482089
$$742$$ 4.87689 0.179036
$$743$$ −32.9848 −1.21010 −0.605048 0.796189i $$-0.706846\pi$$
−0.605048 + 0.796189i $$0.706846\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 45.8617 1.67912
$$747$$ −14.2462 −0.521242
$$748$$ 5.12311 0.187319
$$749$$ 11.3693 0.415426
$$750$$ 0 0
$$751$$ 45.9309 1.67604 0.838021 0.545639i $$-0.183713\pi$$
0.838021 + 0.545639i $$0.183713\pi$$
$$752$$ 17.2614 0.629457
$$753$$ −43.8617 −1.59841
$$754$$ −40.4924 −1.47465
$$755$$ 0 0
$$756$$ 0.630683 0.0229377
$$757$$ −14.6307 −0.531761 −0.265881 0.964006i $$-0.585663\pi$$
−0.265881 + 0.964006i $$0.585663\pi$$
$$758$$ −25.7538 −0.935420
$$759$$ 33.6155 1.22017
$$760$$ 0 0
$$761$$ 31.7538 1.15107 0.575537 0.817776i $$-0.304793\pi$$
0.575537 + 0.817776i $$0.304793\pi$$
$$762$$ 40.9848 1.48472
$$763$$ 17.6847 0.640228
$$764$$ −4.13826 −0.149717
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 18.2462 0.658833
$$768$$ 25.7538 0.929310
$$769$$ −9.50758 −0.342852 −0.171426 0.985197i $$-0.554837\pi$$
−0.171426 + 0.985197i $$0.554837\pi$$
$$770$$ 0 0
$$771$$ −57.6155 −2.07497
$$772$$ 2.35416 0.0847281
$$773$$ −8.06913 −0.290226 −0.145113 0.989415i $$-0.546355\pi$$
−0.145113 + 0.989415i $$0.546355\pi$$
$$774$$ 50.7386 1.82376
$$775$$ 0 0
$$776$$ 36.1080 1.29620
$$777$$ −15.3693 −0.551371
$$778$$ −6.13826 −0.220067
$$779$$ −3.50758 −0.125672
$$780$$ 0 0
$$781$$ 20.4924 0.733277
$$782$$ −36.4924 −1.30497
$$783$$ −8.17708 −0.292225
$$784$$ −4.68466 −0.167309
$$785$$ 0 0
$$786$$ 36.4924 1.30164
$$787$$ 3.82292 0.136272 0.0681362 0.997676i $$-0.478295\pi$$
0.0681362 + 0.997676i $$0.478295\pi$$
$$788$$ 3.12311 0.111256
$$789$$ 54.1080 1.92629
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 22.2462 0.790485
$$793$$ 42.7386 1.51769
$$794$$ 36.6004 1.29890
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 13.0540 0.462396 0.231198 0.972907i $$-0.425736\pi$$
0.231198 + 0.972907i $$0.425736\pi$$
$$798$$ −4.49242 −0.159030
$$799$$ −16.8078 −0.594616
$$800$$ 0 0
$$801$$ 25.3693 0.896381
$$802$$ −42.8466 −1.51297
$$803$$ −10.8769 −0.383837
$$804$$ 7.01515 0.247405
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 73.6155 2.59139
$$808$$ 0.600373 0.0211211
$$809$$ −53.5464 −1.88259 −0.941296 0.337584i $$-0.890390\pi$$
−0.941296 + 0.337584i $$0.890390\pi$$
$$810$$ 0 0
$$811$$ −21.6155 −0.759024 −0.379512 0.925187i $$-0.623908\pi$$
−0.379512 + 0.925187i $$0.623908\pi$$
$$812$$ −2.49242 −0.0874669
$$813$$ −40.9848 −1.43740
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −54.7386 −1.91624
$$817$$ −10.2462 −0.358470
$$818$$ 41.3693 1.44644
$$819$$ −16.2462 −0.567689
$$820$$ 0 0
$$821$$ 40.4233 1.41078 0.705391 0.708818i $$-0.250771\pi$$
0.705391 + 0.708818i $$0.250771\pi$$
$$822$$ −35.5076 −1.23847
$$823$$ 3.50758 0.122266 0.0611332 0.998130i $$-0.480529\pi$$
0.0611332 + 0.998130i $$0.480529\pi$$
$$824$$ −3.50758 −0.122192
$$825$$ 0 0
$$826$$ 6.24621 0.217333
$$827$$ −19.3693 −0.673537 −0.336769 0.941587i $$-0.609334\pi$$
−0.336769 + 0.941587i $$0.609334\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 43.1231 1.49773 0.748864 0.662724i $$-0.230600\pi$$
0.748864 + 0.662724i $$0.230600\pi$$
$$830$$ 0 0
$$831$$ −41.6155 −1.44363
$$832$$ −25.3693 −0.879523
$$833$$ 4.56155 0.158048
$$834$$ 27.5076 0.952510
$$835$$ 0 0
$$836$$ 1.26137 0.0436253
$$837$$ 0 0
$$838$$ −15.2311 −0.526148
$$839$$ −37.1231 −1.28163 −0.640816 0.767695i $$-0.721404\pi$$
−0.640816 + 0.767695i $$0.721404\pi$$
$$840$$ 0 0
$$841$$ 3.31534 0.114322
$$842$$ −15.1231 −0.521177
$$843$$ 42.4233 1.46114
$$844$$ −10.1080 −0.347930
$$845$$ 0 0
$$846$$ 20.4924 0.704544
$$847$$ −4.43845 −0.152507
$$848$$ 14.6307 0.502420
$$849$$ 60.6695 2.08217
$$850$$ 0 0
$$851$$ −30.7386 −1.05371
$$852$$ 8.98485 0.307816
$$853$$ 56.7386 1.94269 0.971347 0.237666i $$-0.0763824\pi$$
0.971347 + 0.237666i $$0.0763824\pi$$
$$854$$ 14.6307 0.500652
$$855$$ 0 0
$$856$$ 27.7235 0.947569
$$857$$ 32.2462 1.10151 0.550755 0.834667i $$-0.314340\pi$$
0.550755 + 0.834667i $$0.314340\pi$$
$$858$$ 46.7386 1.59563
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ −1.26137 −0.0429623
$$863$$ 42.2462 1.43808 0.719039 0.694970i $$-0.244582\pi$$
0.719039 + 0.694970i $$0.244582\pi$$
$$864$$ 3.50758 0.119330
$$865$$ 0 0
$$866$$ −12.8769 −0.437575
$$867$$ 9.75379 0.331256
$$868$$ 0 0
$$869$$ −16.8078 −0.570164
$$870$$ 0 0
$$871$$ −28.4924 −0.965429
$$872$$ 43.1231 1.46033
$$873$$ 52.7386 1.78493
$$874$$ −8.98485 −0.303917
$$875$$ 0 0
$$876$$ −4.76894 −0.161128
$$877$$ 23.7538 0.802108 0.401054 0.916054i $$-0.368644\pi$$
0.401054 + 0.916054i $$0.368644\pi$$
$$878$$ 24.0000 0.809961
$$879$$ −24.8078 −0.836745
$$880$$ 0 0
$$881$$ 45.8617 1.54512 0.772561 0.634941i $$-0.218976\pi$$
0.772561 + 0.634941i $$0.218976\pi$$
$$882$$ −5.56155 −0.187267
$$883$$ −24.4924 −0.824236 −0.412118 0.911131i $$-0.635211\pi$$
−0.412118 + 0.911131i $$0.635211\pi$$
$$884$$ −9.12311 −0.306843
$$885$$ 0 0
$$886$$ −42.7386 −1.43583
$$887$$ 12.4924 0.419454 0.209727 0.977760i $$-0.432742\pi$$
0.209727 + 0.977760i $$0.432742\pi$$
$$888$$ −37.4773 −1.25765
$$889$$ −10.2462 −0.343647
$$890$$ 0 0
$$891$$ −17.9309 −0.600707
$$892$$ 2.87689 0.0963255
$$893$$ −4.13826 −0.138482
$$894$$ 16.9848 0.568058
$$895$$ 0 0
$$896$$ −13.5616 −0.453060
$$897$$ −59.8617 −1.99873
$$898$$ −29.3693 −0.980067
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −14.2462 −0.474610
$$902$$ 12.4924 0.415952
$$903$$ −23.3693 −0.777682
$$904$$ 34.1383 1.13542
$$905$$ 0 0
$$906$$ −87.7235 −2.91442
$$907$$ −50.1080 −1.66381 −0.831904 0.554920i $$-0.812749\pi$$
−0.831904 + 0.554920i $$0.812749\pi$$
$$908$$ −10.3845 −0.344621
$$909$$ 0.876894 0.0290848
$$910$$ 0 0
$$911$$ 4.49242 0.148841 0.0744203 0.997227i $$-0.476289\pi$$
0.0744203 + 0.997227i $$0.476289\pi$$
$$912$$ −13.4773 −0.446277
$$913$$ −10.2462 −0.339100
$$914$$ −13.8617 −0.458506
$$915$$ 0 0
$$916$$ 8.38447 0.277031
$$917$$ −9.12311 −0.301271
$$918$$ −10.2462 −0.338175
$$919$$ −13.3002 −0.438733 −0.219366 0.975643i $$-0.570399\pi$$
−0.219366 + 0.975643i $$0.570399\pi$$
$$920$$ 0 0
$$921$$ 81.1619 2.67438
$$922$$ 7.61553 0.250804
$$923$$ −36.4924 −1.20116
$$924$$ 2.87689 0.0946429
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ −5.12311 −0.168265
$$928$$ −13.8617 −0.455034
$$929$$ −52.1080 −1.70961 −0.854803 0.518952i $$-0.826322\pi$$
−0.854803 + 0.518952i $$0.826322\pi$$
$$930$$ 0 0
$$931$$ 1.12311 0.0368083
$$932$$ 1.36932 0.0448535
$$933$$ −24.6307 −0.806372
$$934$$ 41.4773 1.35718
$$935$$ 0 0
$$936$$ −39.6155 −1.29487
$$937$$ −22.6695 −0.740580 −0.370290 0.928916i $$-0.620742\pi$$
−0.370290 + 0.928916i $$0.620742\pi$$
$$938$$ −9.75379 −0.318472
$$939$$ −80.1771 −2.61648
$$940$$ 0 0
$$941$$ −13.8617 −0.451880 −0.225940 0.974141i $$-0.572545\pi$$
−0.225940 + 0.974141i $$0.572545\pi$$
$$942$$ 15.0152 0.489220
$$943$$ −16.0000 −0.521032
$$944$$ 18.7386 0.609891
$$945$$ 0 0
$$946$$ 36.4924 1.18647
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ −7.36932 −0.239344
$$949$$ 19.3693 0.628755
$$950$$ 0 0
$$951$$ 57.6155 1.86831
$$952$$ 11.1231 0.360502
$$953$$ 24.8769 0.805842 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$954$$ 17.3693 0.562352
$$955$$ 0 0
$$956$$ −0.354162 −0.0114544
$$957$$ −37.3002 −1.20574
$$958$$ −20.4924 −0.662080
$$959$$ 8.87689 0.286650
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −42.7386 −1.37795
$$963$$ 40.4924 1.30485
$$964$$ 5.36932 0.172934
$$965$$ 0 0
$$966$$ −20.4924 −0.659333
$$967$$ 26.8769 0.864303 0.432151 0.901801i $$-0.357755\pi$$
0.432151 + 0.901801i $$0.357755\pi$$
$$968$$ −10.8229 −0.347862
$$969$$ 13.1231 0.421575
$$970$$ 0 0
$$971$$ −49.4773 −1.58780 −0.793901 0.608048i $$-0.791953\pi$$
−0.793901 + 0.608048i $$0.791953\pi$$
$$972$$ −9.75379 −0.312853
$$973$$ −6.87689 −0.220463
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 43.8920 1.40495
$$977$$ 49.2311 1.57504 0.787521 0.616288i $$-0.211364\pi$$
0.787521 + 0.616288i $$0.211364\pi$$
$$978$$ 4.49242 0.143652
$$979$$ 18.2462 0.583151
$$980$$ 0 0
$$981$$ 62.9848 2.01095
$$982$$ −6.52273 −0.208149
$$983$$ 10.4233 0.332451 0.166226 0.986088i $$-0.446842\pi$$
0.166226 + 0.986088i $$0.446842\pi$$
$$984$$ −19.5076 −0.621879
$$985$$ 0 0
$$986$$ 40.4924 1.28954
$$987$$ −9.43845 −0.300429
$$988$$ −2.24621 −0.0714615
$$989$$ −46.7386 −1.48620
$$990$$ 0 0
$$991$$ −20.4924 −0.650963 −0.325482 0.945548i $$-0.605526\pi$$
−0.325482 + 0.945548i $$0.605526\pi$$
$$992$$ 0 0
$$993$$ 30.7386 0.975461
$$994$$ −12.4924 −0.396236
$$995$$ 0 0
$$996$$ −4.49242 −0.142348
$$997$$ −9.68466 −0.306716 −0.153358 0.988171i $$-0.549009\pi$$
−0.153358 + 0.988171i $$0.549009\pi$$
$$998$$ 6.52273 0.206473
$$999$$ −8.63068 −0.273063
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 175.2.a.f.1.1 2
3.2 odd 2 1575.2.a.p.1.2 2
4.3 odd 2 2800.2.a.bi.1.1 2
5.2 odd 4 175.2.b.b.99.2 4
5.3 odd 4 175.2.b.b.99.3 4
5.4 even 2 35.2.a.b.1.2 2
7.6 odd 2 1225.2.a.s.1.1 2
15.2 even 4 1575.2.d.e.1324.3 4
15.8 even 4 1575.2.d.e.1324.2 4
15.14 odd 2 315.2.a.e.1.1 2
20.3 even 4 2800.2.g.t.449.1 4
20.7 even 4 2800.2.g.t.449.4 4
20.19 odd 2 560.2.a.i.1.2 2
35.4 even 6 245.2.e.i.226.1 4
35.9 even 6 245.2.e.i.116.1 4
35.13 even 4 1225.2.b.f.99.3 4
35.19 odd 6 245.2.e.h.116.1 4
35.24 odd 6 245.2.e.h.226.1 4
35.27 even 4 1225.2.b.f.99.2 4
35.34 odd 2 245.2.a.d.1.2 2
40.19 odd 2 2240.2.a.bd.1.1 2
40.29 even 2 2240.2.a.bh.1.2 2
55.54 odd 2 4235.2.a.m.1.1 2
60.59 even 2 5040.2.a.bt.1.2 2
65.64 even 2 5915.2.a.l.1.1 2
105.104 even 2 2205.2.a.x.1.1 2
140.139 even 2 3920.2.a.bs.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 5.4 even 2
175.2.a.f.1.1 2 1.1 even 1 trivial
175.2.b.b.99.2 4 5.2 odd 4
175.2.b.b.99.3 4 5.3 odd 4
245.2.a.d.1.2 2 35.34 odd 2
245.2.e.h.116.1 4 35.19 odd 6
245.2.e.h.226.1 4 35.24 odd 6
245.2.e.i.116.1 4 35.9 even 6
245.2.e.i.226.1 4 35.4 even 6
315.2.a.e.1.1 2 15.14 odd 2
560.2.a.i.1.2 2 20.19 odd 2
1225.2.a.s.1.1 2 7.6 odd 2
1225.2.b.f.99.2 4 35.27 even 4
1225.2.b.f.99.3 4 35.13 even 4
1575.2.a.p.1.2 2 3.2 odd 2
1575.2.d.e.1324.2 4 15.8 even 4
1575.2.d.e.1324.3 4 15.2 even 4
2205.2.a.x.1.1 2 105.104 even 2
2240.2.a.bd.1.1 2 40.19 odd 2
2240.2.a.bh.1.2 2 40.29 even 2
2800.2.a.bi.1.1 2 4.3 odd 2
2800.2.g.t.449.1 4 20.3 even 4
2800.2.g.t.449.4 4 20.7 even 4
3920.2.a.bs.1.1 2 140.139 even 2
4235.2.a.m.1.1 2 55.54 odd 2
5040.2.a.bt.1.2 2 60.59 even 2
5915.2.a.l.1.1 2 65.64 even 2